How to migrate LXD containers from servers with 2.0.11 to others with 5.13?

I don’t think the migration protocol will be compatible between those versions.

I suggest upgrading the source to the snap version (to 4.0), and then you can do lxc export <instance> <filename> transfer the file to the new hosts and do lxc import <filename>.

See Migrate deb 3.0 to snap 4.0 - #2 by tomp